Consistent Supersequences and Transversal Graphs an Extended Abstract

Motivation A consistent supersequence is a common supersequence of the set of positive strings and a common nonsupersequence of the set of negative strings Di erent problems related to consistent supersequences nd applications in molecular biology learning theory data compression manufacturing systems design and draw attention due to their attractive combinatorial structure and challenging complexity aspect Jiang and Li JL were probably the rst to study the complexity of the consistent supersequence existence problem proving that it can be solved in polynomial time with one positive or one negative string and is NP complete with two positive strings Midden dorf M and independently Fraser F complemented their results showing that the problem is NP complete with two negative strings as well In the same paper Midden dorf showed that the shortest consistent supersequence problem is MAX SNP hard even with one negative string and that the longest minimal consistent supersequence problem is strongly NP hard Fraser applied the dynamic programming approach to solve both problems in polynomial time if the number of positive and negative strings is bounded He also noted that the longest consistent supersequence problem with the same constraint remains open However if the number of negative strings is unbounded then as Zhang Z proved the longest consistent supersequence problem is MAX SNP hard even for one positive string Note that we have listed here only intractability results obtained for a binary alphabet A transversal of a given set of strings is a vector where each component is a position in one of the given strings A transversal graph where the set of vertices is the set of all transversals for the set of given strings was introduced by Rubinov and Timkovsky RT RT to reduce nding a longest common nonsupersequence to searching a longest path in the transversal graph They showed that the approach ts also for nding a shortest common supersequence and allows to solve both problems in polynomial time if